What connector attaches to the dome light? Or do I need a new switch? The original wiring doesn't exist for the switches.

The switch pictured is for a 55, 56 chevy . 57's have the spring to hold them in.The switch acts a ground for the dome light so your switch needs to contact bare metal. The pictured switch is rusty and seems to be missing the spring washer(for lack of a better word) that holds it in the hole in the door jam and establishes the ground contact. There are a bunch for sale on ebay.:bowtier: